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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.concepts.state_algebra_operations"></a><a class="link" href="state_algebra_operations.html" title="State Algebra Operations">State
      Algebra Operations</a>
</h3></div></div></div>
<div class="toc"><dl class="toc">
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations">Operations</a></span></dt>
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra">Algebra</a></span></dt>
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.pre_defined_implementations">Pre-Defined
        implementations</a></span></dt>
<dt><span class="section"><a href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.example_expressions">Example
        expressions</a></span></dt>
</dl></div>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
          The following does not apply to implicit steppers like implicit_euler or
          Rosenbrock 4 as there the <code class="computeroutput"><span class="identifier">state_type</span></code>
          can not be changed from <code class="computeroutput"><span class="identifier">ublas</span><span class="special">::</span><span class="identifier">vector</span></code>
          and no algebra/operations are used.
        </p></td></tr>
</table></div>
<h5>
<a name="boost_numeric_odeint.concepts.state_algebra_operations.h0"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.state_algebra_operations.description"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.description">Description</a>
      </h5>
<p>
        The <code class="computeroutput"><span class="identifier">State</span></code>, <code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code>
        together define a concept describing how the mathematical vector operations
        required for the stepper algorithms are performed. The typical vector operation
        done within steppers is
      </p>
<p>
        <span class="emphasis"><em><span class="bold"><strong>y</strong></span> = Σ α<sub>i</sub> <span class="bold"><strong>x<sub>i</sub></strong></span></em></span>.
      </p>
<p>
        The <code class="computeroutput"><span class="identifier">State</span></code> represents the
        state variable of an ODE, usually denoted with <span class="emphasis"><em>x</em></span>. Algorithmically,
        the state is often realized as a <code class="computeroutput"><span class="identifier">vector</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">&gt;</span></code> or <code class="computeroutput"><span class="identifier">array</span><span class="special">&lt;</span> <span class="keyword">double</span> <span class="special">,</span> <span class="identifier">N</span> <span class="special">&gt;</span></code>,
        however, the genericity of odeint enables you to basically use anything as
        a state type. The algorithmic counterpart of such mathematical expressions
        is divided into two parts. First, the <code class="computeroutput"><span class="identifier">Algebra</span></code>
        is used to account for the vector character of the equation. In the case
        of a <code class="computeroutput"><span class="identifier">vector</span></code> as state type
        this means the <code class="computeroutput"><span class="identifier">Algebra</span></code> is
        responsible for iteration over all vector elements. Second, the <code class="computeroutput"><span class="identifier">Operations</span></code> are used to represent the actual
        operation applied to each of the vector elements. So the <code class="computeroutput"><span class="identifier">Algebra</span></code>
        iterates over all elements of the <code class="computeroutput"><span class="identifier">State</span></code>s
        and calls an operation taken from the <code class="computeroutput"><span class="identifier">Operations</span></code>
        for each element. This is where <code class="computeroutput"><span class="identifier">State</span></code>,
        <code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code> have to work together to make
        odeint running. Please have a look at the <code class="computeroutput"><span class="identifier">range_algebra</span></code>
        and <code class="computeroutput"><span class="identifier">default_operations</span></code> to
        see an example how this is implemented.
      </p>
<p>
        In the following we describe how <code class="computeroutput"><span class="identifier">State</span></code>,
        <code class="computeroutput"><span class="identifier">Algebra</span></code> and <code class="computeroutput"><span class="identifier">Operations</span></code> are used together within the
        stepper implementations.
      </p>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.concepts.state_algebra_operations.operations"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations" title="Operations">Operations</a>
</h4></div></div></div>
<h6>
<a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.h0"></a>
          <span class="phrase"><a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.notation"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations.notation">Notation</a>
        </h6>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term"><code class="computeroutput"><span class="identifier">Operations</span></code></span></dt>
<dd><p>
                The operations type
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">Value1</span></code>, ... ,
            <code class="computeroutput"><span class="identifier">ValueN</span></code></span></dt>
<dd><p>
                Types representing the value or time type of stepper
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">Scale</span></code></span></dt>
<dd><p>
                Type of the scale operation
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">scale</span></code></span></dt>
<dd><p>
                Object of type <code class="computeroutput"><span class="identifier">Scale</span></code>
              </p></dd>
<dt><span class="term"><code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code></span></dt>
<dd><p>
                Type that represents a general scale_sum operation, <code class="literal"><span class="emphasis"><em>N</em></span></code>
                should be replaced by a number from 1 to 14.
              </p></dd>
<dt><span class="term"><code class="literal">scale_sum<span class="emphasis"><em>N</em></span></code></span></dt>
<dd><p>
                Object of type <code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code>,
                <code class="literal"><span class="emphasis"><em>N</em></span></code> should be replaced by a
                number from 1 to 14.
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code></span></dt>
<dd><p>
                Type of the scale sum swap operation
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">scale_sum_swap2</span></code></span></dt>
<dd><p>
                Object of type <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">a1</span><span class="special">,</span>
            <span class="identifier">a2</span><span class="special">,</span>
            <span class="special">...</span></code></span></dt>
<dd><p>
                Objects of type <code class="computeroutput"><span class="identifier">Value1</span></code>,
                <code class="computeroutput"><span class="identifier">Value2</span></code>, ...
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">y</span><span class="special">,</span>
            <span class="identifier">x1</span><span class="special">,</span>
            <span class="identifier">x2</span><span class="special">,</span>
            <span class="special">...</span></code></span></dt>
<dd><p>
                Objects of <code class="computeroutput"><span class="identifier">State</span></code>'s
                value type
              </p></dd>
</dl>
</div>
<h6>
<a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.h1"></a>
          <span class="phrase"><a name="boost_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.operations.valid_expressions">Valid
          Expressions</a>
        </h6>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Name
                  </p>
                </th>
<th>
                  <p>
                    Expression
                  </p>
                </th>
<th>
                  <p>
                    Type
                  </p>
                </th>
<th>
                  <p>
                    Semantics
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    Get scale operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Operations</span><span class="special">::</span><span class="identifier">scale</span><span class="special">&lt;</span>
                    <span class="identifier">Value</span> <span class="special">&gt;</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Scale</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Get <code class="computeroutput"><span class="identifier">Scale</span></code> from
                    <code class="computeroutput"><span class="identifier">Operations</span></code>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Scale</span></code> constructor
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Scale</span><span class="special">&lt;</span>
                    <span class="identifier">Value</span> <span class="special">&gt;(</span>
                    <span class="identifier">a</span> <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Scale</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Constructs a <code class="computeroutput"><span class="identifier">Scale</span></code>
                    object
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Scale</span></code> operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">scale</span><span class="special">(</span>
                    <span class="identifier">x</span> <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="keyword">void</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Calculates <code class="computeroutput"><span class="identifier">x</span> <span class="special">*=</span> <span class="identifier">a</span></code>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Get general <code class="computeroutput"><span class="identifier">scale_sum</span></code>
                    operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="literal">Operations::scale_sum<span class="emphasis"><em>N</em></span>&lt; Value1
                    , ... , ValueN &gt;</code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code>
                  </p>
                </td>
<td>
                  <p>
                    Get the <code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code> type
                    from <code class="computeroutput"><span class="identifier">Operations</span></code>,
                    <code class="literal"><span class="emphasis"><em>N</em></span></code> should be replaced
                    by a number from 1 to 14.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">scale_sum</span></code> constructor
                  </p>
                </td>
<td>
                  <p>
                    <code class="literal">ScaleSum<span class="emphasis"><em>N</em></span>&lt; Value1 , ... , ValueN
                    &gt;( a1 , ... , aN )</code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="literal">ScaleSum<span class="emphasis"><em>N</em></span></code>
                  </p>
                </td>
<td>
                  <p>
                    Constructs a <code class="computeroutput"><span class="identifier">scale_sum</span></code>
                    object given <code class="literal"><span class="emphasis"><em>N</em></span></code> parameter
                    values with <code class="literal"><span class="emphasis"><em>N</em></span></code> between
                    1 and 14.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">scale_sum</span></code> operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="literal">scale_sum<span class="emphasis"><em>N</em></span>( y , x1 , ... , xN )</code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="keyword">void</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Calculates <code class="computeroutput"><span class="identifier">y</span> <span class="special">=</span> <span class="identifier">a1</span><span class="special">*</span><span class="identifier">x1</span>
                    <span class="special">+</span> <span class="identifier">a2</span><span class="special">*</span><span class="identifier">x2</span>
                    <span class="special">+</span> <span class="special">...</span>
                    <span class="special">+</span> <span class="identifier">aN</span><span class="special">*</span><span class="identifier">xN</span></code>.
                    Note that this is an <code class="literal"><span class="emphasis"><em>N</em></span>+1</code>-ary
                    function call.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Get scale sum swap operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">Operations</span><span class="special">::</span><span class="identifier">scale_sum_swap2</span><span class="special">&lt;</span>
                    <span class="identifier">Value1</span> <span class="special">,</span>
                    <span class="identifier">Value2</span> <span class="special">&gt;</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Get scale sum swap from operations
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
                    constructor
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span><span class="special">&lt;</span> <span class="identifier">Value1</span>
                    <span class="special">,</span> <span class="identifier">Value2</span>
                    <span class="special">&gt;(</span> <span class="identifier">a1</span>
                    <span class="special">,</span> <span class="identifier">a2</span>
                    <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Constructor
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">ScaleSumSwap2</span></code>
                    operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">scale_sum_swap2</span><span class="special">(</span> <span class="identifier">x1</span>
                    <span class="special">,</span> <span class="identifier">x2</span>
                    <span class="special">,</span> <span class="identifier">x3</span>
                    <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="keyword">void</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Calculates <code class="computeroutput"><span class="identifier">tmp</span> <span class="special">=</span> <span class="identifier">x1</span></code>,
                    <code class="computeroutput"><span class="identifier">x1</span> <span class="special">=</span>
                    <span class="identifier">a1</span><span class="special">*</span><span class="identifier">x2</span> <span class="special">+</span>
                    <span class="identifier">a2</span><span class="special">*</span><span class="identifier">x3</span></code> and <code class="computeroutput"><span class="identifier">x2</span>
                    <span class="special">=</span> <span class="identifier">tmp</span></code>.
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra" title="Algebra">Algebra</a>
</h4></div></div></div>
<h6>
<a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.h0"></a>
          <span class="phrase"><a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.notation"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra.notation">Notation</a>
        </h6>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term"><code class="computeroutput"><span class="identifier">State</span></code></span></dt>
<dd><p>
                The state type
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">Algebra</span></code></span></dt>
<dd><p>
                The algebra type
              </p></dd>
<dt><span class="term"><code class="literal">Operation<span class="emphasis"><em>N</em></span></code></span></dt>
<dd><p>
                An <code class="literal"><span class="emphasis"><em>N</em></span></code>-ary operation type,
                <code class="literal"><span class="emphasis"><em>N</em></span></code> should be a number from
                1 to 14.
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">algebra</span></code></span></dt>
<dd><p>
                Object of type <code class="computeroutput"><span class="identifier">Algebra</span></code>
              </p></dd>
<dt><span class="term"><code class="literal">operation<span class="emphasis"><em>N</em></span></code></span></dt>
<dd><p>
                Object of type <code class="literal">Operation<span class="emphasis"><em>N</em></span></code>
              </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">y</span><span class="special">,</span>
            <span class="identifier">x1</span><span class="special">,</span>
            <span class="identifier">x2</span><span class="special">,</span>
            <span class="special">...</span></code></span></dt>
<dd><p>
                Objects of type <code class="computeroutput"><span class="identifier">State</span></code>
              </p></dd>
</dl>
</div>
<h6>
<a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.h1"></a>
          <span class="phrase"><a name="boost_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions"></a></span><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.algebra.valid_expressions">Valid
          Expressions</a>
        </h6>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Name
                  </p>
                </th>
<th>
                  <p>
                    Expression
                  </p>
                </th>
<th>
                  <p>
                    Type
                  </p>
                </th>
<th>
                  <p>
                    Semantics
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    Vector Operation with arity 2
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">algebra</span><span class="special">.</span><span class="identifier">for_each2</span><span class="special">(</span>
                    <span class="identifier">y</span> <span class="special">,</span>
                    <span class="identifier">x</span> <span class="special">,</span>
                    <span class="identifier">operation2</span> <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    void
                  </p>
                </td>
<td>
                  <p>
                    Calls <code class="computeroutput"><span class="identifier">operation2</span><span class="special">(</span> <span class="identifier">y_i</span>
                    <span class="special">,</span> <span class="identifier">x_i</span>
                    <span class="special">)</span></code> for each element <code class="computeroutput"><span class="identifier">y_i</span></code> of <code class="computeroutput"><span class="identifier">y</span></code>
                    and <code class="computeroutput"><span class="identifier">x_i</span></code> of <code class="computeroutput"><span class="identifier">x</span></code>.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Vector Operation with arity 3
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">algebra</span><span class="special">.</span><span class="identifier">for_each3</span><span class="special">(</span>
                    <span class="identifier">y</span> <span class="special">,</span>
                    <span class="identifier">x1</span> <span class="special">,</span>
                    <span class="identifier">x2</span> <span class="special">,</span>
                    <span class="identifier">operation3</span> <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    void
                  </p>
                </td>
<td>
                  <p>
                    Calls <code class="computeroutput"><span class="identifier">operation3</span><span class="special">(</span> <span class="identifier">y_i</span>
                    <span class="special">,</span> <span class="identifier">x1_i</span>
                    <span class="special">,</span> <span class="identifier">x2_i</span>
                    <span class="special">)</span></code> for each element <code class="computeroutput"><span class="identifier">y_i</span></code> of <code class="computeroutput"><span class="identifier">y</span></code>
                    and <code class="computeroutput"><span class="identifier">x1_i</span></code> of
                    <code class="computeroutput"><span class="identifier">x1</span></code> and <code class="computeroutput"><span class="identifier">x2_i</span></code> of <code class="computeroutput"><span class="identifier">x2</span></code>.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Vector Operation with arity <code class="literal"><span class="emphasis"><em>N</em></span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="literal">algebra.for_each<span class="emphasis"><em>N</em></span>( y , x1 , ...
                    , xN , operation<span class="emphasis"><em>N</em></span> )</code>
                  </p>
                </td>
<td>
                  <p>
                    void
                  </p>
                </td>
<td>
                  <p>
                    Calls <code class="literal">operation<span class="emphasis"><em>N</em></span>( y_i , x1_i ,
                    ... , xN_i )</code> for each element <code class="computeroutput"><span class="identifier">y_i</span></code>
                    of <code class="computeroutput"><span class="identifier">y</span></code> and <code class="computeroutput"><span class="identifier">x1_i</span></code> of <code class="computeroutput"><span class="identifier">x1</span></code>
                    and so on. <code class="literal"><span class="emphasis"><em>N</em></span></code> should be
                    replaced by a number between 1 and 14.
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.concepts.state_algebra_operations.pre_defined_implementations"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.pre_defined_implementations" title="Pre-Defined implementations">Pre-Defined
        implementations</a>
</h4></div></div></div>
<p>
          As standard configuration odeint uses the <code class="computeroutput"><span class="identifier">range_algebra</span></code>
          and <code class="computeroutput"><span class="identifier">default_operations</span></code>
          which suffices most situations. However, a few more possibilities exist
          either to gain better performance or to ensure interoperability with other
          libraries. In the following we list the existing <code class="computeroutput"><span class="identifier">Algebra</span></code>/<code class="computeroutput"><span class="identifier">Operations</span></code> configurations that can be
          used in the steppers.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    <code class="computeroutput"><span class="identifier">State</span></code>
                  </p>
                </th>
<th>
                  <p>
                    <code class="computeroutput"><span class="identifier">Algebra</span></code>
                  </p>
                </th>
<th>
                  <p>
                    <code class="computeroutput"><span class="identifier">Operations</span></code>
                  </p>
                </th>
<th>
                  <p>
                    Remarks
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    Anything supporting <a href="http://www.boost.org/doc/libs/release/libs/range/" target="_top">Boost.Range</a>,
                    like <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">vector</span></code>, <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">list</span></code>,
                    <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code>,... based on a <code class="computeroutput"><span class="identifier">value_type</span></code> that supports operators
                    +,* (typically <code class="computeroutput"><span class="keyword">double</span></code>)
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">range_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">default_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Standard implementation, applicable for most typical situations.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code> based on a <code class="computeroutput"><span class="identifier">value_type</span></code> that supports operators
                    +,*
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">array_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">default_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Special implementation for boost::array with better performance
                    than <code class="computeroutput"><span class="identifier">range_algebra</span></code>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Anything that defines operators + within itself and * with scalar
                    (Mathematically spoken, anything that is a vector space).
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">default_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    For the use of <a class="link" href="controlled_stepper.html" title="Controlled Stepper">Controlled
                    Stepper</a>, the template <code class="computeroutput"><span class="identifier">vector_space_reduce</span></code>
                    has to be instantiated.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">device_vector</span></code>, <code class="computeroutput"><span class="identifier">thrust</span><span class="special">::</span><span class="identifier">host_vector</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">thrust_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">thrust_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    For running odeint on CUDA devices by using <a href="http://code.google.com/p/thrust/" target="_top">Thrust</a>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    Any RandomAccessRange
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">openmp_range_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">default_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    OpenMP-parallelised range algebra
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">openmp_state</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">openmp_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">default_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    OpenMP-parallelised algebra for split data
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">array</span></code> or anything which allocates
                    the elements in a C-like manner
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">vector_space_algebra</span></code>
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">mkl_operations</span></code>
                  </p>
                </td>
<td>
                  <p>
                    Using the <a href="http://software.intel.com/en-us/articles/intel-mkl/" target="_top">Intel
                    Math Kernel Library</a> in odeint for maximum performance.
                    Currently, only the RK4 stepper is supported.
                  </p>
                </td>
</tr>
</tbody>
</table></div>
</div>
<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="boost_numeric_odeint.concepts.state_algebra_operations.example_expressions"></a><a class="link" href="state_algebra_operations.html#boost_numeric_odeint.concepts.state_algebra_operations.example_expressions" title="Example expressions">Example
        expressions</a>
</h4></div></div></div>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Name
                  </p>
                </th>
<th>
                  <p>
                    Expression
                  </p>
                </th>
<th>
                  <p>
                    Type
                  </p>
                </th>
<th>
                  <p>
                    Semantics
                  </p>
                </th>
</tr></thead>
<tbody><tr>
<td>
                  <p>
                    Vector operation
                  </p>
                </td>
<td>
                  <p>
                    <code class="computeroutput"><span class="identifier">algebra</span><span class="special">.</span><span class="identifier">for_each3</span><span class="special">(</span>
                    <span class="identifier">y</span> <span class="special">,</span>
                    <span class="identifier">x1</span> <span class="special">,</span>
                    <span class="identifier">x2</span> <span class="special">,</span>
                    <span class="identifier">Operations</span><span class="special">::</span><span class="identifier">scale_sum2</span><span class="special">&lt;</span>
                    <span class="identifier">Value1</span> <span class="special">,</span>
                    <span class="identifier">Value2</span> <span class="special">&gt;(</span>
                    <span class="identifier">a1</span> <span class="special">,</span>
                    <span class="identifier">a2</span> <span class="special">)</span>
                    <span class="special">)</span></code>
                  </p>
                </td>
<td>
                  <p>
                    void
                  </p>
                </td>
<td>
                  <p>
                    Calculates <span class="emphasis"><em><span class="bold"><strong>y</strong></span> = a1
                    <span class="bold"><strong>x1</strong></span> + a2 <span class="bold"><strong>x2</strong></span></em></span>
                  </p>
                </td>
</tr></tbody>
</table></div>
</div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2009-2015 Karsten Ahnert and Mario Mulansky<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
</tr></table>
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